Editing Compact space (section)

==== Ordered spaces ====

For an ordered space {{math|(''X'', <)}} (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
# {{math|(''X'', <)}} is compact.
# Every subset of {{mvar|X}} has a supremum (i.e. a least upper bound) in {{mvar|X}}.
# Every subset of {{mvar|X}} has an infimum (i.e. a greatest lower bound) in {{mvar|X}}.
# Every nonempty closed subset of {{mvar|X}} has a maximum and a minimum element.

An ordered space satisfying (any one of) these conditions is called a complete lattice.

In addition, the following are equivalent for all ordered spaces {{math|(''X'', <)}}, and (assuming [[countable choice]]) are true whenever {{math|(''X'', <)}} is compact. (The converse in general fails if {{math|(''X'', <)}} is not also metrizable.):
# Every sequence in {{math|(''X'', <)}} has a subsequence that converges in {{math|(''X'', <)}}.
# Every monotone increasing sequence in {{mvar|X}} converges to a unique limit in {{mvar|X}}.
# Every monotone decreasing sequence in {{mvar|X}} converges to a unique limit in {{mvar|X}}.
# Every decreasing nested sequence of nonempty closed subsets {{mvar|S}}<sub>1</sub> ⊇ {{mvar|S}}<sub>2</sub> ⊇ ... in {{math|(''X'', <)}} has a nonempty intersection.
# Every increasing nested sequence of proper open subsets {{mvar|S}}<sub>1</sub> ⊆ {{mvar|S}}<sub>2</sub> ⊆ ... in {{math|(''X'', <)}} fails to cover {{mvar|X}}.